In this paper, structural properties of chordal graphs are studied, establishing a relationship between these structures and integer Laplacian eigenvalues. We present the characterization with equal vertex algebraic connectivities, by means vertices that compose minimal separators graph; we stablish sufficient condition for cardinality maximal clique to appear as an eigenvalue. Finally, review ...

let $n$ be any positive integer and let $f_n$ be the friendship (or dutch windmill) graph with $2n+1$ vertices and $3n$ edges. here we study graphs with the same adjacency spectrum as the $f_n$. two graphs are called cospectral if the eigenvalues multiset of their adjacency matrices are the same. let $g$ be a graph cospectral with $f_n$. here we prove that if $g$ has no cycle of length $4$ or $...

In a recent issue of this MONTHLY, Hetzel, Liew, and Morrison [4] pose a rather natural question: what is the probability that a random n× n integer matrix is diagonalizable over the field of rational numbers? Since there is no uniform probability distribution on Z, we need to exercise some care in interpreting this question. Specifically, for an integer k ≥ 1, let Ik = {−k,−k + 1, . . . , k − ...

The undirected power graph of a finite group $G$, $P(G)$, is a graph with the group elements of $G$ as vertices and two vertices are adjacent if and only if one of them is a power of the other. Let $A$ be an adjacency matrix of $P(G)$. An eigenvalue $lambda$ of $A$ is a main eigenvalue if the eigenspace $epsilon(lambda)$ has an eigenvector $X$ such that $X^{t}jjneq 0$, where $jj$ is the all-one...

We revisit Hoffman relation involving chromatic number χ and eigenvalues. We construct some graphs and weighted graphs such that the largest and smallest eigenvalues λ dan μ satisfy λ = (1 − χ)μ. We study in particular the eigenvalues of the integer simplex T 2 m, a 3-chromatic graph on ( m+2 2 ) vertices.